## Operations Research - Dec 2014

### Mechanical Engineering (Semester 7)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.

### Attempt the following: (Any Four)

**1 (a)** State the general rules for convening any primal LPP into its dual.(5 marks)
**1 (b)** Give Johnson's procedure for determining an optimal sequence for processing jobs on three machines.(5 marks)
**1 (c)** Explain different behaviour of server and customers in a queue.(5 marks)
**1 (d)** Mention the reasons for using simulation for solving real-life problems.(5 marks)
**1 (e)** Give the Classification of different Inventory Models(5 marks)
**2 (a)** A company is manufacturing two products A and B. the net profit for these products is Rs.60 and Rs.50 respectively. These products require working in two departments C and D. The available hours per month in these departments are 150 each. Product A requires 2 hours in department C and 3 hours in department D. Product B requires 3 hours in department C and 2 hours in department D. the production of the products A and B cannot exceed 40 units each because of marketability constraints. Formulate the LP. model and solve it by simplex method.(12 marks)
**2 (b)** The Taj Service Station has five machines each of whom can service a scooter in 2 hours on the average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at the service station at an average rate of 2 scooters per hour. Assuming that the scooter arrivals are Poisson distributed and the servicing times
are distributed exponentially, determine:

a) Utilization factor

b) The probability that the system shall be idle(8 marks)
**3 (a)** Use two phase simplex method to solve the problem.

Minimize Z= x_{1}-2x_{2}-x_{3}

Subject to -2x_{2}+3x_{2}+3x_{3}=2

2x_{1}+3x_{2}+4x_{3}=1

x_{1},x_{2},x_{3}≥0(10 marks)
**3 (b)** Solve the travelling salesman problem given by the following
data:

C_{12}=20, C_{13}=4, C_{4}=10,
C_{23}=5,C_{34<./sub>=6, C25=10, C35=6,C45=20,
where Cij=Cji and there is no route between cities I and J if the
value for Cij is not shown}(10 marks)
**4 (a)** Use Big M method to solve the following LPP:

Maximum Z=3x+8y

Subject to x+y=200

y≥60

x≤80

x,y≥0(10 marks)
**4 (b)** A large computer installation contains 2000 components of identical nature which are subject to failure as per probability distribution that follows:

Month end | 1 | 2 | 3 | 4 | 5 |

% failure to date | 10 | 25 | 50 | 80 | 100 |

Components which fail have to be replaced for efficient functioning of the system. If they are replaced as and when failures occur. the cost of replacement per unit is Rs 3 Alternatively. If all components are replaced in one lot at periodic intervals and individually replace only such failures as occur between group replacements, the cost of component replaced is Re 1.

Assess which policy of replacement would be economical(10 marks)

**5 (a)**A steel company has three open hearth finances and [we rolling mills. Transportation cost (rupees per quintal) for shipping steel from timers to rolling mills are shown in the following table: What is the optimal shipping schedule?

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Furnaces | Mills | Capacity (in quintal) | ||||

M1 | M2 | M3 | M4 | M5 | ||

F1 | 4 | 2 | 3 | 2 | 6 | 8 |

F2 | 5 | 4 | 5 | 2 | 1 | 12 |

F3 | 6 | 5 | 4 | 7 | 3 | 14 |

Requirement(in quintal) | 4 | 4 | 6 | 8 | 8 |

**5 (b)**Explain the principle of dominance and hence solve the following game

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Player 'B' | ||||

Player 'A' | 1 | 2 | 3 | |

I | 8 | 5 | 8 | |

II | 8 | 6 | 5 | |

III | 7 | 4 | 5 | |

IV | 6 | 5 | 6 |

**6 (a)**The Paints ltd would like. to improve its inventory management policies for its supply of paints used for automobiles. Annual demand for such paint is 50,000 litres and the paint which costs Rs. 20 per litre. is used at a constant rate. Annual carrying cost are estimated at 15 percent of the value of paint held. Each order costs Rs.80. Determine:

(i) How much paint should be ordered each time?

(ii) How often should paint be ordered?

(iii) Time between two consecutive orders.

(iv) What is the total annual cost associated with this policy?(10 marks)

**6 (b)**A bakery keeps stock of popular brand of bread. Previous experience indicates the daily demand as given below:

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Daily Demand |
0 | 10 | 20 | 30 | 40 | 50 |

Probability |
0.01 | 0.2 | 0.15 | 0.5 | 0.12 | 0.02 |

Consider the following sequence of random numbers:

48,78,19,51,77,15,14,68,9

Using above sequence. simulate the demand for next 10 days.

(i) Find out the stock situation if the ovner of the bakery decides to make 30 breads every day.

(ii) Estimate the daily average demand for the bread on the basis of simulated data.(10 marks)

**7 (a)**Use dynamic programming to find the shortest path from Bombay to Calcutta along lines joining various vertices lying between Bombay and Calcutta. Length of each path is given

img(10 marks)

**7 (b)**A company has a demand of 12,000 writs/year for an item and it can produce 2,000 such items per month. The cost of one setup is Rs.400 and the holding cost/units/month is Rs. 0.15. Find the optimum lot size and the total cost per year, assuming the cost of one unit as Rs. 4. Also find the maximum inventory, manufacturing time and total time.(10 marks)